34 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
Q - Q 0 9 
Q 7T 
— _ ( r l2 + r J3 + r l4 + r 23 + r 24 + r 3i) + ( ~ j ( r 14 r 23 + r U V 34 + ^ 13 ^ 24 ) 
+ (>’ 12 3 + ?'i3 3 + r n 3 + ^ 23 3 + Lu 3 + Ln) + ) (V l3 r u + r ls 
’l2 r 23 r 24 
/ 2 \3 1 
+ r l3 r^r 3i + r ]4 r 24 r 34 ) + ( -) jg- (r 14 s r 23 + r 14 r 23 3 + r 13 r 24 3 + r 1:i V 21 + r 13 V M 
+ V 34 3 ) + 
2 \2 1 
tt / |T ( r l2 2r l4 r 23 + r l3 3r l4 r 23 + r n r \i V U + V 14^24 + T V^J r U 
+ r l4’’23 r 24 2 + r l^Ai + V l<l r 14^34 + V^ r Z4 + + r n r,Jr^ 
+ r l 3 r 24 r 34 2 ) +.(lxXXvii.). 
This is the correct value including terms of the fourth order, but to this order of 
approximation we can throw it into a much simpler form. Let r ss , = sin S ss >, then 
<l> “ 77 ^" \ — sin -1 r 12 + sin -1 r 13 + sin -1 r 14 -f sin -1 r 23 + sin -1 r 24 + sin -1 r 34 
Oo 
+ — (sin 1 r 12 sin 1 r ]3 sin 1 r u -f sin 1 r 12 sin 1 r . 23 sin 1 r 24 
+ sin -1 r 13 sin -1 r 23 sin -1 r 34 + sin -1 r 14 sin -1 r 24 sin -1 r 34 ) 
+ ~ [sin" 1 r 14 sin" 1 r 23 {(l - r 12 2 ) (1 - r 13 2 ) (1 - r 24 2 ) (1 - r 34 2 )}-" 
+ sin' 1 r 12 sin -1 r 34 {(l - r 13 2 ) (1 - ?’ 14 2 ) (1 - r.J) (1 - r 2i 2 )} 4 
+ sin" 1 r 13 sin" 1 r 24 {(l - r 12 2 ) (1 - r 14 2 ) (1 - r 23 2 ) (1 - r 3 ^)}~ h ] 
= S ]2 + S 13 + §n + S 23 + S 24 + §34 
+ ~ (§12§13§14 + §12^23§24 + §13§23§34 + §14§24^34.) 
, 2 /S 14 S 23 cos 8 ]4 cos S 23 + S P S 34 cos S 10 cos 8 34 + S 13 8 04 cos 8 n cos 8 04 
+ ~ 
7r 
cos 8 13 cos S 13 cos S 14 cos S 23 cos S 24 cos 8, 
34 
We may write this 
where 
7 r 
2 
U2 
U3 
• Q — Q 0 7T 
TP / 
sm — - 
v vn 
= cos E 
— §14. — §23 — 
^24 §34 
(lxxxviii.). 
(lxxxix.) 
— Tj- (§i2§j3^i4 ~b S 12 8 23 S 24 + S 13 S 23 S 24 -f S 14 8 24 8 34 ) 
2 /'S u So, cos S,, cos 8,, + 8, 9 8o d cos S 19 cos So, + S 13 S 2 , cos S 13 cos 8, 
7 r 
cos 8]o cos 8 ns cos S 14 cos 8 0S cos S gi cos 8 : 
J 13 
14 uuo u 23 uuo u 24 wo o 34 
The expressions E and E' of (lxxxvi.) and (lxxxix.) are of considerable interest, for 
they enable us to express the area of a spherical triangle in three-dimensioned space, 
