408 REPORT— 1883. 
so that the first term in the series which enters is of the order + In fact, we 
have after some easy reductions 
log x!=log V 2m +4 (w +4) log (a? +24+2)—(« +3) 
oho by lig 4 Secuee 
24023 1602t 453602" * 
and therefore when we write 
8 Ren cnn 
alaVigi Veer a a 
the error is less than 
u : 1 Kani 2 
sana of the oe ze. less than 0.2 of the error in adopt 
ing the ordinary expression. 
6. On a Generalised Hypergeometric Series. By Professor 
A. R. Forsyru. 
The object of the communication was to deduce for the series 
a.a+1.8.84+1.0.6+1 2, <a 
Coy Oa & Diente ual 
14%. 
€ 
the relations corresponding to those which hold for Gauss’s series as they occur in 
the memoirs of Gauss (‘Ges, Werke,’ bd. iii.) and Kummer (‘Crelle,’ t. XV.) 
7. Note on a Simple Method of Solving the General Equation of the 
Fourth Degree. By Aurrep Lovee. 
If in the =" (ay, a, dy, a, a4) (x, 1)*=0, the substitution y =a, + a, be made, 
the resulting =” is of the form 
y'+6Hy’+4Gy+F=0 . : : ie (LL))5 
which can be arranged as the difference of two squares, viz. 
2 
(y? + 8H + 29°)? (2-7) = Wie. ee ae 
by introducing a quantity g, which only occurs in the absolute term of the equa- 
tion and is determined by the cubic equation in 9’, 
49°+12Hg+(9H?-F)g—-G=0 . . . (8), 
If, then, we can obtain one root of this cubic (and there is always at least one 
positive root), the solution of the biquadratic is complete. 
By substituting a, =? + H, the cubic can be reduced to the form 
4p°-Ipt+J=0 . : ; j . (4, 
where 
I(=a,a,—4a,a, + 3a,*), 
and 
I(=| MB UN, A ) 
bi) Ongitls 
a are 
are the two invariants of the original biquadratic, whose invariance gives us 
a, I= F+ 3H", and a3J=a2HI- G?-4H3, 
_ [It may be noticed as an interesting fact that each term of these expressions 
involves the coefficients of the original equation to an order equal to the weight. ] 
* Published in eatenso in the Quarterly Journal of Mathematics, vol. xix. pp. 292-337. 
