304 
PROFESSOR G. H. DARWIN ON THE PEAR-SHAPED FIGURE 
Also 
- {3, 2} {3, 3} 
q-i 8 - ^o- - 6^ 
i + I/3), 
aiul 
Whence on substitution 
= .AD with qJ = 1. 
qz n-i 
P.3M-) 
(1 - 
_ 3 - /3 + 2R 3\ 
1 - A/ V 5(1 - /3) ,/ 
( 6 ). 
.S' = 3 ; OOS, (z.) = ^q^ Pgi (i^) + ^3 (z/), with r^g = I. 
The equation for yScr is 
_ - 2} (3, 3| 
8 - /Str + 6/3 ■ 
We may derive the result from the last case by introducing 
(iJg)^ “ 1 + f/3 (1 + jS), 
and changing the sign of/3, so that 
/3cr = 4 (I + — i^g), 
^ ^ = 1- 
'i3 P 
Whence on substitution 
i/(r) = ry, + /3 - £ 3 ) - 1). 
(7). 
and 
The forms of the corresponding functions of g are the same, excejDt that (I — 
1 + /3 
1 -yS 
— replace the corresponding fg-ctors. 
I have not determined the cosine- and sine-functions, because they may be written 
down at once from the I'esults already obtained. The three roots of the fundamental 
1 —/3 cos2(/) 
cubic are \j?, and 
1-/S 
Hence we have only to re23lace v~ by this last 
function in the seven formulm (1)—(7) in order to obtain functions proqDortional to 
the seven cosine- and sine-functions. If the definition of the latter functions is to 
agree with that given in “ Harmonics,” the factors must be determined appropriately, 
but the question as to the value of the factor will not arise here. 
