824 
ME. J. W. L. GLAISHER ON RICCATI’S 
(iv.) 1874. Bach. 
£ Annales Scientifiques 
pp. 47-68. 
“ De 1’Integration paries Series de 1’Equation —-—- 
d.e l’Ecole Normale Superieure.’ Deuxieme serie, vol. iii., 
Detailed account, with developments, of (i.) and (iii.). In (iii.) n is written in place 
1 . X r L 
of - and /3 in place of —, so that the series are reduced to the forms given in art. 16. 
If the differential equation is similarly transformed it becomes 
dhc n — 1 clu Q A 
Ji¥ ‘ 
This is the form of the equation adopted by M. Bach, who finally deduces the series 
in the case of Biccati’s equation. The form is a very convenient one. See art. 16. 
(v.) 1878. Glajsher. “Example Illustrative of a Point in the Solution of 
Differential Equations in Series.” ‘ Messenger of Mathematics,’ vol. viii, pp. 20-23. 
In the well-known expansions quoted in art. 11, viz. 
< ] _ y ( i-4*)p= ii'xA 1 
{1 -f v / ( 1 — 4a:) y = 2 p i 1 —px ^ ^ x~ —1— —pp—— x s -f &c. 
L - ; ^ ; 
the series are such that if p is an integer, one terminates, and after a certain number 
of zero terms, recommences and reproduces the other. It follows therefore that the 
differential equation whose general integral is 
u=c r(l — ^/(l — 4r)p'-|-c 2 { 1 + y( 1 — 4a) y 
must afford an example of the principle pointed out in (i.). The differential equation 
is found to be 
~ {(¥-6)a-^-h 1 }~-p(p-l)u=0, 
and its integration in series affords the illustration referred to in the title. The note 
was suggested by art. 11. See art. 15. 
(vi.) 1878. - “ Generalised Form of Certain Series.” ‘ Proceedings of the 
London Mathematical Society,’ vol. ix., pp. 197-202. 
Theorems deduced from 
