PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 7 



rr d £ ds 



. "J dn 

 V tf= limit of - , (1) 



the integration extending throughout the surface of T, of which dS is an element. In fact, if 

 I, m, n be the direction-cosines of the normal, we shall have 



"dU ,„ rrf,dU dU dU 



ff—dS- ff(l~ + m~~ n—-)dS 

 J J dn J J \ dx dy dz) 



-IJfjy d%+ IS d ^ dxdx+ lS d TJ a,dy (2)> 



We have also, supposing the origin of co-ordinates to be at the point P, as we may without 

 loss of generality, 



— = (— ] + ( — - ] x + [- — - ) y + [- — — ] * + terms of the 2nd order, &c. (3), 

 dx \dxj \dx 1 \dxdyj \dxd«/ 



where the parentheses denote that the differential coefficients which are enclosed in them have 



rrdU 

 the values which belong to the point P. In the integral -r-dy dz, each element must 



J J dx 



be taken positively or negatively, according as the normal which relates to it makes an acute 



or an obtuse angle with the positive direction of the axis of w. If we combine in pairs the 



elements of the integral which relate to opposite elements of the surface of T, we must write 



II — — - - — — dydz, where the single and double accents subscribed refer respectively to 



the first and second points in which the surface of T is cut by an indefinite straight line 

 drawn parallel to the axis of x, and in the positive direction, through the point (o, y, z). 

 We thus get by means of (3), omitting the terms of a higher order than the first, which vanish 

 in the limit, 



rrfdU dll\ , (d*U\ rr 



Uw: - -ix") ** *• = u$ ifa - •-) * **> 



But ff(x /t - x t ) dy dz is simply the volume T. Treating in the same manner the two other 

 integrals which appear on the right-hand side of equation (2), we get 



Dividing by T and passing to the limit, and omitting the parentheses, which are now no longer 

 necessary, we obtain the theorem enunciated. 



If in equation (l) we take for T the elementary volume r 2 sin 6 drd9d(p, or rdrdOdz, 

 according as we wish to employ polar co-ordinates, or one of three rectangular co-ordinates 

 combined with polar co-ordinates in the plane of the two others, we may at once form the 

 expression for yd, and thus pass from rectangular co-ordinates to either of these systems 

 without the trouble of the transformation of co-ordinates in the ordinary way. 



