442 MATHEMATICAL APPENDICES 45 



while that carried over in the other direction is 



Q 2 = dydz(m/4:ir)NT- l \ X dx'\* dy'^ ^e^Vr 



J _oo J 00 J 00 



The difference between these magnitudes 



or the sum of the gain and loss of one-half, is the friction exerted 

 from the side of increasing x upon the other ; and so 



denotes the reaction exerted by the half corresponding to the 

 smaller values of x on the half with the larger values of x. 



Since, as above assumed, v is a continuous function of x, y, z, 

 and therefore also v' a continuous function of a/, y r , z f , Taylor's 

 theorem gives the development 



* = v + (*-*)* + (y'-y)~ +('-) + ... 



After substitution of this series the integrations can be carried out, 

 and present no difficulty if the rectilinear coordinates are replaced 

 by polar coordinates whose origin is at the point (x, y, z), i.e. by 

 the coordinates r and s already introduced, and a second angle $ 

 given by 



(a/ x) = r cos s 



(y r y) r sm s c s 



(z f z) = r sin s sin 0, 



where the sign must be determined so that the acute angle s may 

 satisfy these relations. Then we obtain 



cos s sin s dr ds d 



Ci = dy dz(m/7r)NT- i r f" (V* 

 Jo Jo Jo 



C 2 = dy dz(ml7r)NT- 1 (^ (^ f M e-^vj cos s sin s dr ds <fy, 

 Jo Jo Jo 



where for shortness we put 



v/ = t> + os s + sin s cos < + 



1\^ dy \ dz 



vj = v - \(^ cos s+ ^ sin s cos <f> + ~ sins sin 0V + . . . 

 l\dx dy dz Y J 



Eemembering that v and its differential coefficients are independent 



