206 PROFESSOR A. E. H. LOVE ON THE 



member of the equation (94) will be found to be j80i/(24i/ 19), which is negative 

 when f > v > 0. Hence the left-hand member of this equation is negative when 

 x = 0. Also it will be found by the method of asymptotic expansion (cf. 22 above) 

 that, when x is great, the left-hand member is approximately equal to 1280aT 4 for 

 all values of v. After the previous cases, in which the corresponding equations have 

 a single root, we are led to expect that in this case there is no root, for it is unlikely 

 that there is more than one. We proceed to verify this expectation in the cases 

 where v = and v = . 



31. Multiply the left-hand member of (94) by a; 4 and put v = 0. We get 



8\ (3x t +25x 2 + 2Q) - 

 L 



= 0, 



or, since 10 is a factor, 



9x t +20x a -105-(9x*+Ux 4 -l5x?-l05)x- l e-^{' ! e l '*dx = 0. 



Jo 



The term of lowest degree in the left-hand member, when expanded in powers of 

 x, is - 3 fi a; 8 ; when x is great, the left-hand member approximates to 128. Now 

 multiply by xe**, and put 



y = (Qx i +20x 3 -l05x)e^-(9x 6 +llx 4 -15x a -l05)\ Z e^dx. 



Jo 



We know that when x is small y is small and negative of the order x 7 , and that 

 when x is great y is great and negative of the order 128x6**. Now 



[* 



and, if we put z for x~ l dy/dx, 



z = (54a?-3Qx)e**- (54z 4 + 44z 2 -30) f' 



Jo 



where z is negative both when x is small and when x is great ; also 



and if we put w for x~ l dz/dx, 



w = 88a:e' I -(216z 3 +88) F tfdx, 



Jo 

 where w is negative both when x is small and when x is great ; and now 



</./; 



