398 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Multiplying symbolically, we have thirty-two different combinations. The 
first sixteen will be each unique, but the second half 
[ 
/ 
/ 
/ 
f " 
1 
V , + 
P 5 + 
P> + 
V , 
I 
I 
i 
With a little practice all the special forms existing may be easily worked out. 
The general theorems on which this paper is based are given in the 
following lemmas. These in the form in which they are shown, so far as I 
know, are new, but they are so simple that it seems unlikely they have not 
been proved many times before. They depend on the well-known formulae 
for the transference of the moments of a body round the centre of gravity 
to a point at a distance h therefrom, and on the values of the moments of 
a point binomial round the centre of gravity. If /x 2 , ^ 3 , yu 4 be the moments 
of a curve round the centre of gravity, and // 2 , jul' 3 , /ul\ be the moments 
round a point at a distance h, then 
fl O = /x 2 + h 2 , 
/x 3 = /x 3 + 3/qx 2 + Ti ® , (A) 
fj\ = p 4 + 47&/X, 3 + 6 A 2 /x 2 + IP . 
Also the moments of (^ + l) n about the centre of gravity are given by 
c 2 nq 
^“(2 + 15*’ 
Pz ~ 
c 2 nq(q — 1) 
w- d :; ’ 
Pnq 
3(»-2)g ) 
(?+l) 2 /' 
Lemma I . — If an expression of distributed terms be multiplied by a 
second expression of distributed terms, the moments of the compound 
expression round the centroid vertical are given by 
p2 = £ 2 "t £ 2 ’ 
^3 = £ 3 £ 3 ’ 
/*4 = f 4 + *£”2 J 
where f' 2 , f" 2 , etc., are the moments of the separate expressions, and /x 2 , 
etc., those of the compound expression. Let the second expression be 
denoted by a + b + c + . . . ; then the centre of gravity of the compound 
expression is moved a distance equivalent to the distance of the middle 
point of the new expression from its centre of gravity. For the first 
can be conceived as concentrated at its centre of gravity, and if its mass 
