1901-2.] Dr Muir on the Theory of Orthogonants. 
273 
is headed “Problema I.” and occupies §§ 10-15, pp. 264-279, its 
subject being an algebraical transformation pure and simple. The 
third and longest is headed “Problema II.” and concerns the 
closely related, not to say dependent, problem of the transformation 
of a double integral. With this clear-cut subdivision there is no 
need for any process of sifting : we turn at once to Problema I. 
It is stated by Jacobi as follows : — Proponitur, per substitutiones 
Uneares 
X = as 
+ 
as 
+ 
a!' s' 
w 
= at 
+ 
bu 
+ cv 
CO 
CQ. 
II 
+ 
ft's' 
+ 
Ff 
w' 
ji 
+ 
b'u 
+ cv 
«Sl 
II 
Co 
+ 
r 's' 
+ 
ys 
w" 
So 
ii 
+ 
b''u 
+ cv 
quae identice efficiant 
2 2 2 ' 2 2 2 
x + y + z = s + s' + s " , 
2 ,2 ,/2 2 2 2 
10 +10 +10 = t + U 4 - V , 
transformare expressionem 
(Ax + By + C z)w + (Ax + B 'y + C 'z)id + (A'x + B "y + C "z)w" 
in hanc simpliciorem 
G st + G 's 21 + G "s"v . 
Among the problems of the previous papers its closest relative is the 
first of all, the relation being that of general to particular. In 
modern symbolism the expression now given for transformation is 
X 
y 
z 
A 
B 
C 
w 
A 
B' 
C' 
id 
A" 
B" 
C" 
id 
er of 1827 
it is 
X 
V 
z 
A 
B 
C 
X 
A' 
B' 
C' 
y 
A" 
B" 
C" 
z . 
Cauchy’s extension was from one set of three or four variables to 
one set of n variables; Jacobi’s from one set of three variables to 
two sets of three variables. 
The preparation for solution begins with the reminder that the 
condition 
PROC. ROY. SOC. EDIN. — VOL. XXIY. 
18 
