[645 
645] a smith’s prize paper, 1877. 41 
uniform density, on a 
,, , . A cos(2w+l)a? . . 
that is, 0 , the value m question gives sin (2n + 1) x = 1, and therefore 
in the equation of a 
;rsection of two given 
cos(2n q-1)x = 0 ; and it does not give cos£P = 0; hence every such factor 1 Sma 
sin a 
is a double factor, or we have 
conics passing through 
se, show how to obtain 
the equations of the 
1 — sin (2n + 1) x = (1 + sin x) II (\ — S ^- æ ) . 
\ sin a) 
Or the like result might be obtained by considering instead of 1 — sin (2n-t- V)x t 
the more general function 
of a ternary quadric 
sin (2n + 1) a ± sin (2n q-1) x, 
rential equation, given 
r are the differential 
ndent variables x, y, z 
and finally assuming a=\ir. 
3. Relates to a theory which is not, but ought to be, treated of in the text 
books of the University. See Serret’s Algèbre Supérieure, t. n., Sect. iv. 
ns, but only to notice 
The substitutions which leave ab + cd unaltered are 
iven integral function 
he given function is 
ional, the most simple 
1 1, that is, the letters remain unchanged, 
a (ab), that is, a and b are interchanged, 
/3 (cd), that is, c and d are interchanged, 
(x) = 0, here dividing 
7 (ab) (cd), that is, a and b and also c and d are simultaneously interchanged, 
d integral function of 
; that is, 
8 (ac) (bd), same with a and c, b and d, 
e (ad) (be), same with a and d, b and c, 
f (acbd), that is, we cyclically change a into c, c into b, b into d, and d into a, 
ve thus —-— in the 
x — a 
, 1 
expression of . 
x — a 
factional function, and 
of x which does not 
can be expressed as a 
6 (adbc), that is, we cyclically change a into d, d into b, b into c, and c into a, 
viz. we have eight substitutions 1, a, /3, 7, 8, e, Ç, 6 forming a group; that is, the 
product of any two of them, in either order, is a substitution of the group (or, 
what is the same thing, the effect of the successive performance of the two upon 
any arrangement abed is the same as that of the performance thereon of some other 
substitution of the group); thus we have a 2 = l, /3 2 = 1, 7 2 =1, a/3 =/3a = 7, &c. ; the 
system of these equations, which verify that the set of substitutions form a group, 
defines the constitution of the group—thus to take a more simple instance, a group 
function of sin x, of 
tains, as is at once 
the factor (1 q- sin x). 
of 4 may be 1, a, a 2 , a 3 (a 4 = l) or 1, a, /3, a/3, (a 2 = l, /3 2 = 1, a/8 =/3a). 
4. The expression of the general coefficient is 
1.2 ... 2 n 
1; then this will be 
~~ 1.2...?i — a.l .2 ... n + a’ 
which can be transformed by the well-known formula 
1.2 ...n = n n+ \*/(tt) e n > 
C. X. 
6