PEOFESSOE C. J. JOEY ON QUATEENIONS AND PEOJECTIVE GEOMETEY. 2G5 
We projDOse to obtain functions from given functions J\, /g, &c., which fall 
into certain classes connected by invariantal relations. We denote two arbitrary 
functions by the Koman capitals X, Y, and we consider the transformations 
effected by multiplying a given function by X and into Y. 
This transformation changes the series of functions 
... A, /i. /,,■■■ /lAA.. • • . (243) 
into the series 
X./VY, x/,Y. xy^Y,... X/JVVW. ■ • • x/i./yy^/rVA . (24 4); 
and we shall speak of this as the (XY) class. 
I nP QPTIPQ 
/r‘. ff\ ff\ ■ ■ ■ ■ ■ ■ /rA/r'/i/r* • • • ( 245 ) 
becomes 
Y-/r‘x-', y->/3->x->. ... Y-y.-'/OAX-’, ... 
Y-'/ry/s-VtA-'X-*.(240), 
and this is the (Y"\ X~^) class. 
The S3ries 
fiff\ fJi', — /i/r'A/r’.(247) 
is the (XX“^) class, transforming into the series 
x/,y-'x-', xyy-'x->,...x/,y-y,/r‘x-'. . . . (248); 
and finally the series 
/r'A /r'A ■ • ■ /rA/r'A.( 249 ) 
forms the (Y“'Y) class, as it transforms into 
Y-'/r‘AY, Y-'/r'yY,... Y->/rAA-'AY . . . (250). 
Inverse functions of the (XY) class belong to the (Y“^X”^) class, and conversely; 
inverse functions of the classes (XXor (Y“ W) belong to their own class, and so 
also do products and cjuotients of functions of these classes. The product of an 
(XY) function into a (Y“^X"^) function is an (XX~^) function, and so on. 
In like manner there are four classes for the conjugate functions, as appears on 
taking the conjugates of a typical function. The annexed scheme exhibits the eight 
classes, the conjugates being printed under their correspondents : — 
(XY), (Y-iX-i) (XX-i), (Y-iY) 
(Y'X') (X'-W'-i) (X'-iX') (Y'Y'-i).(251). 
60. When we deal with quadrics or complexes, or when the condition is imposed 
that self-conjugate functions remain self-conjugate, the classes of the conjugate type 
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