82 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
c-i -f- C.2 =- a, 
( 71^1 H" 73%) ~ 
[7i%i (1 + Uf) + 72%2 (1 + w/)} air = fi^ah^ 
{yi% (1 + + 72% (1 + 3^)1 
[7,%j (i + -1- Su,^) + 73 "% (1 + 6w/ + 31^2^)] ah* = iji.ah*, 
<7i% + 73®% (1 + 10^'3^ + 15M]^)} aP = fx,aP. 
The first equation here represents tlie equality of the areas of the resultant curve and 
its components. Reducing to the simplest terms, we have the following six equations 
to find the six unknowns, z-^, z.^, 71 , 72 , % • — 
% + % = 1.(8). 
7i^i + 73% = 0.(0). 
7i'^'i (1 + '*h~) + 73% (1 + '^h~) = p3.(10)- 
7 j^^i (1 + 3Ri~) + 72% (1 + 3%") — fxo .( 11 ). 
7i^^i (1 “t“ Oi/q” + ^Ui) + 72% (1 + 0^3^ d" 32 ^ 2 ”^) == P'4. . . . (12). 
7,%i(1 + 10%+15O + 73®%(1 + 10«3" + 15%^) = /^5 • • (13). 
Equations (8)-(13) give the complete solution of the problem.* After several trials, 
I find that the elimination of z-^^, z^, Ui, u.^ from these equations, and the determination 
of equations giving 7^73 and 7 ^ + y., appear to lead to a resulting equation of the 
lowest possible order. 
(7.) Eliminating 2:3 between ( 8 ) and (9), we have 
Similarly, 
72 
7 i - 72 
(14). 
23 
( 15 ). 
* All my attempts to obtain a simpler set have failed. Equating of selected ordinates, or of selected 
portions of area, or of moments round the axis of x, all appear to lead to exponential equations defying 
solution. It is possible, however, that some other six equations of a less complex kind may vxltimately 
be found. 
