282 PKOFESSOR G. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
represents a surface. We shall write generally for any differential 
dQ = ?nSp dg-.(353), 
where is a homogeneous quaternion function of q and of the order m—1. Since p 
is a determinate function of q, q may be regarded as a function of p ; and using 
Euler’s theorem for homogeneous functions we have 
Q z= ^pq = P.(354) 
where P is the function of p into which Q transforms. 
86. Again, we shall write generally for the differential of the quaternion p regarded 
as a function of q, 
dp = (?n — l)f dq.(355) 
whei'e fdq is a linear function of dq, involving q homogeneously in the order m — 2. 
This function is self-conjugate, for taking two successive and independent differen¬ 
tials of Q, 
d' dQ = 7?iSp d' dq -p m {m — 1) S . f d'q . dq 
= dd'Q = m'^p dd'r^' + {m — l) B . fdq . d'q.(356) ; 
and because the differentials are independent, 
d'dq = dd'q, and therefore Sdqfd'q = Sd'qfdq .... (357), 
consequently the function/is self-conjugate, for dq and d'q are quite arbitrary. 
87. Differentiating (354) we find on comparison Avith (353) 
dP = uSqdp, where (u — 1) (m — 1) = 1 . . . . (358), 
and it is easy to verify that n is the order in Avhich p is involved in P. Also 
Introducing a new linear function g, we Write 
dq = [71 — l)q dp.(359), 
and, as in the last article, g is self-conjugate and immlvesp in the order 7 i — 2 in its 
constitution. 
Thus for any differential by (355) and (359) 
dp = (?n — l)/’dq = (m — 1) (r — l)/.q dp =fg dp . . . (360); 
or symbolically 
1 =./i/ = f7/.(361)’ 
and one function ])roduces on an arbitrary quaternion the same effect as the inverse 
of the other. In particular, employing Euler’s theorem in (355) and (359) Ave have 
P =f9. = 9~\ ’ 9 = 9P =f~^P .(362)- 
