182 MR. ROBERT RAWSON ON SINGULAR 



solution, as may be inferred from (30) by expanding 



x 

 cos . - and taking c = i. 



The condition #=o gives ~ = infinity and -j-l - ) = o. 



Boole says, " If -j- = infinity leads to a solution which 



does not involve (y) in its expression, nothing is to be 

 inferred whether it is singular or not. Then the proper 



test is -r-( -J = mfinityv" Why not? the inference here 



is that x=o is a particular solution of (29), which has 

 been already proved. 



The equation y=o does not satisfy (29), as may be seen 

 from the following development : ; — 



Since 



. X 



ajsm- 



y 



y z 



= — cosec 



X 



X 



y 











X X.X 



X 



by 



3 6 °y 3 



+ &C. 



.} 





_ y* + y + 7^ 



x z 6 360?/ 



■+ 3 

 15 



IX* 



\2oy l 



+ &C. 







= infinity. 











icsinoo 





Therefore y^ = infinity in (29) when y=o. 



9. The following particular cases of (21) and (22) are 

 interesting. 

 Let 



f{w)=\ogw, .-. /'(w)=i. 



