;320 PEOFESSOR C. J. JOEY OX QUATERXIOXS AXD PROJECTIVE GEOMETRY. 
153. When lines are to be converted into lines, the conditions are 
yi^-'i + y't^>t 
: — 
+ . . . 
• (557), 
and tlie array 
r./pb f'/h ■ ■ 
./Aft 
h\ ft 0 . . 
.00 ft ft ] 
! ./f' 1 ./d' 1 • • 
yydi ft 
0 l\ . . 
. ft 0 ft ft 1 
^1 
./Aft^ ft 
. 
1 
r • 
(558) 
j ./f-ft" y^ft" • • 
ft ft () . . 
. l>n h',, ft ft j 
l /irt'/" .. 
0 
0 ft ft . . 
. ft ft h,„ V,„ J 
of n “h 2 m columns and of 
2 m row’s must vanisli. 
The number of conditions is 
Gift — n + 1 . Thus a function of a seven-system and of a thirteen-system 
respectively converts one and two lines into one and two others. 
In like manner, when planes are to be converted into planes, the array is of 
11 + oni columns and of rows, and recpiires 9»i — -T 1 conditions for its 
vanishing. 
In general for space of p dimensions a function of an n-system is completely 
defined if 
11 — fX (^^1 0)11^ -|- &C.) d~ 1 — jU.j\I 1 . . . . ( 559 ), 
which converts 1 n^ given points, rn., lines, m. planes, &c., into other given points, 
lines and planes, &c. 
154. We sliall now suppose that the array (556) does not vanish without conditions 
restricting the generality of the points. Let all the points except ii„i and he given. 
It is sufiicient to consider the cases in wdiich the number of conditions does not 
exceed three. 
By the expansion (55ft) we have, if fftn — -j- I = n, so that v conditions must be 
satisfied, or if n = 3??i — v I = 3 [m — 1) + (4 — a), 
2 ± (/l («j),/:2('b)’/3(«l). (.f’4.(^L)>/5(«-)’/G ('k)> • • • 
(./sw-j — ^ —])> S 3'"-3 1 ) ^5«_j) 
^ <.y 3 '«—i ('-ft")’y 3'"—1 (^ 6 ")' • • y's '"—"+1 (^ft")’ — o • . • (56o). 
For it is obviou.sly no use retaining any term (/] (<"^ 1 ),/i: (fft), ^3 («'i), y ’4 («i)), in 
wdilch a h does not enter, as the minor array of this term has a column of zeros and 
vanishes. 
We thus have three types of conditions for n = 1 , 2 or 3, and these are of the 
forms, the functions F being linear. 
1. (FiO,„, Fn4„„ Fgre,,, />„) = 0 ; 
II. [FjO,,,, Iba,„, = ft ; 
III. {Fpe,,, h,„] =0 .(5G1). 
