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## Geometric theory of Fresnel diffraction patterns

### Proceedings of the Indian Academy of Sciences - Section A (1945-04-01) 21: 177-187 , April 01, 1945

### Summary

The paper deals with a new approach to the subject of diffraction which enables the nature of the Fresnel pattern due to an obstacle or aperture of arbitrary form to be easily derived. Taking the hint suggested by observation, it is assumed that in the region of shadow the pattern is produced by the interference of radiations having their origin in the boundary of the aperture or the obstacle. In the region of light, the boundary radiation interferes with the primary incident waves and produces the fluctuations of intensity. The radiation from the boundary can again be effectively replaced by spherical waves originating from a finite number of point-sources on the boundary, called ‘poles’, at which the path to the point of observation from the boundary is a maximum or a minimum. This immediately gives a geometric definition of the poles as those points on the boundary whose projection on the plane of observation are the feet of the normals from the observation point to the projection of the boundary. Using this result, it is possible to geometrically map out the positions of maximum and minimum intensity in the diffraction pattern of an arbitrary aperture or obstacle. The relation between the Fresnel and Fraunhofer patterns of an aperture is discussed and it is shown that the Fresnel pattern approaches more and more to the other type as the size of the aperture is diminished or the distance to the observation screen is increased.

## The lower order of the zeros of an integral function (II)

### Proceedings of the Indian Academy of Sciences - Section A (1945-04-01) 21: 162-164 , April 01, 1945

## A note on the chemical composition of castor leaves

### Proceedings of the Indian Academy of Sciences - Section A (1945-03-01) 21: 123-125 , March 01, 1945

### Summary

Ricinine has been shown to be the only definite crystalline component of castor leaves. An easy method of isolating it from this convenient source is described. It is only feebly bitter and mildly toxic.

## Dynamics of quadrupedal locomotion

### The bulletin of mathematical biophysics (1945-06-01) 7: 53-57 , June 01, 1945

O. Fischer's dynamical equations for unbranched systems in two dimensions are extended to the branched systems exhibited by a quadruped.

## Studies in inorgano-organic gels in pinene

### Proceedings of the Indian Academy of Sciences - Section A (1945-03-01) 21: 105-113 , March 01, 1945

### Summary

The phenomenon of syneresis exhibited by gels of sodium oleate in pinene has been systematically investigated. The velocity of syneresis increases as (i) temperature is lowered and (ii) concentration is decreased. The nature of the synereticum has been examined. A theory has been suggested which explains the effects of several factors on the kinetics of syneresis of these gels. The law which holds for the imbibition of liquids is also applicable to the phenomenon of syneresis. Also, the kinetics of syneresis of these gels obey an equation which is similar to that of an inhibited reaction.

## A contribution to the mathematical biophysics of visual aesthetics

### The bulletin of mathematical biophysics (1945-03-01) 7: 41-45 , March 01, 1945

In continuation of previous studies, inequalities between different parameters of the brain are derived which determine whether an individual prefers in general visual patterns consisting of a relatively small number of relatively strongly excited elements, or such patterns which consist of a very large number of relatively weakly excited elements. As has been discussed in a previous publication, the first type of pattern is usually represented by artificial human-made designs, whereas the second type of pattern is formed predominantly in natural landscapes and sceneries. Thus the inequalities established in this paper give us the biophysical conditions which determine an individual's preference for either artificial designs or for landscapes and other natural objects.

## Pauli’s identities in the Dirac algebra

### Proceedings of the Indian Academy of Sciences - Section A (1945-12-01) 22: 408-422 , December 01, 1945

### Summary

It is shown in this paper that by choosing suitable forms for 4×4 matrices as products of Dirac matrices and matrices of rank unity (*i.e.*, products of 4×1 and 1×4 matrices), and expressing them as linear combinations of the sixteen elements*γ*^{A} of the basis of the Dirac algebra, one can derive the generalized identities of Pauli which hold in this algebra. Generalizations are given for cases not dealt with by Pauli, and the use of his B-matrix is also avoided. The same method yields further ‘tensor’, multilinear, and polynomial identities of which it is shown that the last two kinds of identities are derivable from bilinear and quadratic ones. It is pointed out that all types of identities can be deduced by considering five primitive types of matrices.

## Condensation of aldehydes with malonic acid

### Proceedings of the Indian Academy of Sciences - Section A (1945-12-01) 22: 400-404 , December 01, 1945

### Summary

The condensation of*p*-tolylaldehyde with malonic acid by the pyridinetrace method gave, on repetition, a nearly quantitative yield of the*p*-tolylideneacrylic (*p*-methylcinnamic) acid.*p*-Tolylidenemalonic acid has been isolated for the first time in this condensation, in the absence of pyridine, in 75% yield.

*m*-Tolyladehyde, on condensation with malonic acid in the presence of pyridine in traces, gave up to 68% of the*m*-methylcinnamic acid. By a longtime-room-temperature condensation in the presence of both pyridine and piperidine, the yield came up to the theoretical. The*m*-tolylidenemalonic acid came out best in the presence of glacial acetic acid, but the yield was not great, being =48%.

## Flächenstreifen

### Vorlesungen Über Differentialgeometrie I (1945-01-01) 1: 67-84 , January 01, 1945

### Zusammenfassung

Die aus einem Punkte *x* und einer hindurchgehenden Ebene *s* bestehende geometrische Figur wollen wir als ein *Flächenelement* bezeichnen. Für die Anschauung ist es zweckmäßig, sich bei einem Flächenelement von der Ebene *e* immer nur ein kleines Stück in der Umgebung des Punktes *x* vorzustellen. Zu der Ebene *e* gehören zwei entgegengesetzt gerichtete, zu ihr senkrechte Einheitsvektoren, die Einheitsvektoren der *Normalen des Flächenelements*. Zeichnen wir einen der beiden Vektoren aus, so wird dadurch eine positive Seite des Flächenelements festgelegt, nämlich die Seite, nach welcher der Vektor hinzeigt. Das Flächenelement wird, wie wir sagen wollen, *gerichtet*. Durch Angabe des Punktes *x* und des Normalenvektors *ξ* in ihm ist dann das gerichtete Flächenelement eindeutig festgelegt. Zu jedem regulären Punkt einer Fläche gehört ein Flächenelement, das durch den Flächenpunkt und die durch ihn hindurchgehende *Tangentenebene* (§41) der Fläche gebildet wird. Als Tangentenebene eines Flächenpunktes bezeichnen wir dabei die Ebene, die durch alle Tangenten an die von ihm auslaufenden Flächenkurven aufgespannt wird. Geben wir allgemein *x* und *ξ* als Funktionen eines Parameters *t*, so erhalten wir eine Schar von Flächenelementen.