20 GANESHPRASAD, 



Case II : 



Therefore x'^'^^ r< \-X{x') \ -< x' 



hence l \-X{x') \ l(x'). 



X' (x') 1 — 

 Therefore r\j— and, consequently, dr\j\Jt. 



X {^X ) X 0 



Therefore 



J V (x, t)\^- X(\/Ö, i. e. ~y . (1) 



Thus, when 7p{x')^~, it follows from (1) that W (x, t)\-^l i( Ic, + v>l. 



"When ■tp{x')r^~ or k + v = 1, \V' {x, t)\^l. In all other cases (1) leaves it 

 uncertain whether 



|F'(^,^)|^gl. 



Summary relating to the behaviour of V (x, t) for t small. 



19. The results obtained in Arts. 11 — 18 may be summed up as follows : 

 i. \V'{x, t)\o^l if \B(x, x')\r^l. 



In particular \\V' [x, t)\ — A] -•^l if \\B {x, x')\ — A \ -^1 , A being a finite 

 positive quantity. 



iL \V' (x,t)\>^l{i \D{x, x')\>^l. 



iii. When D {x, x') x' cos 1 1/; {x') { , {x') being 1 and 0 < Zc < 1, 

 V{x,t)±::^l, \r{x,t)\^l, or \r{x,t)\-^ ^]]\ 

 according as ^ {x') ^ l >- ^, or >- Z but ^ ^ ■ 



Behaviour of V{x, t) for t small. 



20. As V{x, t) is an even function of x it would be sufficient to consider 

 the case 0 < a; < 



Proceeding as in Art. 11, it is found without difficulty that 



V{x, t) = ~ f fix') & (x' + x) dx' + ^ rf(x') 0 ix' - x) dx'. 

 It is easily found , by the method of Art. 12 , that as t approaches zero 



