( 365 ) 
of the sum which according to this condition must be a minimum: 
Widor (sdk ee oe + (9m — Im)" + (9g—9m)? 
gp represents the rate preceding S, and g, the rate following Sg: 
If to this quantity we add: 
Zhn (9, + Ia al gien sia is + 9m—1 + gn) 
their sum U will also be a minimum for the same values g, to gyn, 
i=m . 
i=l 
These values g,.-.gm are found by means of the condition that 
each derivative of U taken with reference to each variable shall be 
zero. At the same time we must assign a definite value to 4, in 
i=m 
order to satisfy the equation gj = mQn. 
i 
Thus we obtain the following equations: 
Ip HIN —I + hn = 9 | 
Tee Te dende pd 
Up alten Germ Ob = cts oe he ee 
mm + 2 Iml — Im + Kp 
Onl -- 2 In — Iq + bn 0" 
Bij taking the sum of these m equations, each multiplied by 7 (m—1), 
we eliminate 9, ,4i, gm, the coefficient of each gi of that sum being 
i=m 
equal to 2; hence the sum of the terms is 2 3 g:= 2 mQn. 
=! 
The coefficient of k is: 
>: ety m (m-+-1) (m 4-2) 
gl Uk ve € 
Hence the sum of the multiplied equations is: 
1 2 
nn EO OE, m (m+ —_ ) HEA 
_ 6p + 99 — 2 Qn) 
mt 1) (m+ 2) 
Then we determine the values of g; by multiplying the m equa- 
tions by the terms of the following series: 
mil), 2(m—i+1),...¢@—1)(m—i+1), i(m—i+1), i(m—i),..., 42, 71 
whence km 
