116 



expressions of which the sum is unity, as it ought to be. We may 

 remark that the former of these two expressions represents the square 

 of the cosine, and the latter the square of the sine, of the angle 

 which a near ray makes with the axis of revolution of the system. 



Other Method of obtaining the Approximate Integral. 



8. Again, the approximate integral (C") of the partial differen- 

 tial equation (P')j may be obtained in the following manner. Since 

 V is supposed capable of being developed according to positive inte- 

 ger powers of a, let us assume 



V 



^ 



= FCO) -t- FCi) « 4- F(2) „2 + F(3) «3 ^ (jj„j 



neglecting rf", and considering F^°\ F'*', F'^', F'^', as functions of z, of 

 which the forms are to be determined. To determine these forms, 

 we have, when ;j = 0, 



dV .„.. dW , _„, d^V 



(1") 



iS 



di -^ IT ' <^ ~ '^ dz ' dvi^dz - ^ dz ' dn'dz - ^ dz 



dV dV 



The equation (P') shews that -^ = ± [/,, when n = :, and-^ 

 is positive, if we suppose the motion of the light directed from the 

 negative towards the positive part of the axis of » ; we have there- 

 fore, by (/"), 



dz 



-=1. (K") 



The equation (P') gives also, by differentiating it with respect 

 to >?, 



