( 106 ) 



1 



Tlie ierms nefi^lected in these expressions for p and — are of the 



a 



3'^ order with respect to tlie intervals of time. 



I shall now proceed to show how we can avail onrselves of these 



values for p and — for tlie calculation of the ratios of the triangles. 

 a 



In mv pre\ious paper I have demonstrated tliat the area of 

 the triangle PZF^ considered as a function of r = k {t — ^i) satisfies 

 the differential equation i'^-]- c 7'"":= 0. The same differential equation 

 is satisfied by the area of the triangle P^ZP, considered as a 

 function of r=zh[t^ — t). The two areas may, according to Mac 

 Lauhix be expressed in series of the ascending powers of t. If 

 the variable t takes the vaUie k {i^ — t^) = Tj, the two triangles become 

 equal to J\Z1\ ; therefore it will be possible to obtain a new expansion 

 in series for double the area £\ P^ZF^, by putting in the sum of the 

 two foruier series TrrrTg. From this new series we can easily 

 remove the terms with the even powers of r. 



According to this plan I give here first some higher derivatives 

 of the function F, expressed in F, F, z and derivatives of z with 

 respect to the same variable. 



/^"i = — : F — zF 



F^' = (4^.^ - ci") F 4- {z' - 3H) F 



F^'i =(--=' + \r- + Izz — c^^') F+ 2 i^zz — -Iz^^') F 



/'vii _( )/^_|_ (I3,g _|_ lOi^ _ 5c^^' — z') F. 



triangle PZP, 



For T=:0, the value of the function ; = F\k{t—t,)]=l (t) 



VP 



and that of its first derivative is known, viz. i^j = and i^„ ^ -|" I • 

 The above mentioned expansion in series for LPZP^ is therefore : 



LPZP, It 1 t' . t^ , 1 



\/p 





AP ZP 



The function — = G [k (^2 — 0] = ^^^ (0 i^»f' its derivative also 



y/p 



take for t =z t^ or t = the values 6^^ = and 6^^, =1 -j- |, and so 



