510 REPORT—1862. 
If F be compounded of f, f,...f,, and a single transformation of F into 
t,xf.x..f, be given, we may, by the same principles, find all the trans- 
formations of F into the product of f, f,, f,, taken asin the given transforma- 
tion. Forif F(X,, Y,)=lfrepresent the given transformation, and F(X, Y) 
=IIf be any other transformation, we find X=aX,+6Y,, Y=yX,+éY,, 
ao—Py=+1, and consequently F (4«X,+Y,, yX,+oy,)=F (X,, Y,); or 
a, 2 
78 
is, by Lemma 4, a proper automorphic of F. The formula X=aX, 
+fY,, Y=yX,+8Y,, in which oH 
represent all the transformations Zrteioe 
If F be transformable into f, xf, x..f,, and ® contain F, while f,, f,,..f,, 
contain 9,,%,,-+,, ® will be transformable into ¢,x@,.-¢,- This follows 
from a preceding general observation (Art. 105); but we must add here that 
if T, +; denote positive or negative units, according as the transformations of 
® into F, and f, into ¢, are proper or improper, while v; denotes a positive or 
negative unit according as f,is taken directly or inversely, ¢,; will be taken 
directly or inversely according as T x7; x v; is positive or negative. This is 
is an automorphic of F, will therefore 
apparent if we observe that the sign of the quantity “- vee ‘is altered by an im- 
proper transformation of X, Y, or w;, y;, but is not pete by a transformation 
of any of the other sets. 
The theorem that “forms compounded of equivalent forms, similarly taken, 
are themselves equivalent” is included in the preceding. We may, there= 
fore, speak of the class compounded of any number of given classes. 
It is an important and not a self-evident proposition, that if F be com- 
pounded of ¢, f,, f,,--fn, and be compounded of f,, f,, F is compounded of 
Fis Fare -Fne Let p=at°+26n+yn’, let » be the greatest common divisor 
of a, 23, y, and vy the determinant of ¢; let also X, Y transform F into 
oxf,xf,x..xf,. Writing in X and Y for and » the bipartite expressions 
linear in w, y,, v, y,, by which ¢ is transformed into f, x f,, we obtain a trans- 
formation of F into f,xf,x..xjfn. Ifk be the greatest common divisor of the 
determinants of the matrix of this transformation, Dk* is the greatest common 
d; 
divisor of the n numbers — 3 me IIm?. But this common divisor is the same as 
1=n 
the greatest common divisor of yx II m?;, and the n—2 numbers 
i=3 : 
s=n 
kad II m,? 4=3 CNG 
me 33 
because v is the greatest common divisor of d, m,? and d, m,’ (4th condlu- 
sion), and because p =m, m, (5th conclusion) ; +. ¢., ‘Die= D, or k?= 1, and Fis 
compounded of f,, f,,..f,. Also, if 1 >2, f, is similarly taken in both coms 
Aifi Ai fi 
rT rt i “Oxh X.+Xf, 
_aXdY¥ en dX dY dé dn dé dn IS 
i= da, Ty, Uys da, —\ dé dn ~ dy x) (zr dy, dy, a) Wes ziy 
Q and w; be positive or negative units, according as g and f, are taken directly 
positions, for are identical; and if i=1, or 
