112 



REPORTS ON THE STATE OF SCIENCE. — ^1916. 



§ 11. We can now find the linear differential equation solved by X. 

 By eliminating V and its derivates from 



X X 



X™+?X"=2(V^ + 1V) 

 2V 



X X 



X" + -Xi = 



X 



2X=V" + *'Vi 



X 



we obtain a;''.Xi^ + 4a;3.X™+a;2X"-a;X'-4a;^X=0. 



The corresponding equations for the V, Z, and W functions are : 

 x'Y^'' + 4:X^V^"-3x'W+3xV-ix'Y =0. 



a;*Z'^ + 4a;'Z^"-8a;2Z"-3a;Zi + Z(3-4a;'') =0. 

 x'Wl + 4a;' W"^ + a;' W" + xW- W(l + 4^'') =0. 



§ 12. These equations afford the best means of determining the 

 coefficients in the expansions in series of the functions. The results are 

 set out here ; the appropriate solution is of course determined by multi- 

 plying out a few terms of the expansions of her x, ker x, &c. 



^^^''^~ 11 12 ^[r^4^l2l3l6 + l3T^ 



w%)=eA)+|^3+|^+f^;+ 



z«(x)=^z%)-(|j+^2:|p^ +(2:6^:0)^+ .. 



