DETERMIXATION OF THl-; THEIIMAT, CONDUC'I'I VITY OF MAK'BLE. 271 



X(i\v ii' \vc ik'tennino the vnlues i)f h, lor different valiie.'5 of time, 

 the a'siind ß's might be determined. As tht' series is n rii})idly eon- 

 verging one, we may stop at 4tli nv 5th term, and then with a large 

 numhi'r of (h'termination of" «„ for different timea, we may determine 

 the values of the a's, and ß's by the method of least squares. There 

 is however annlher, and far simpler method, which in this particular 

 ease, at least is no less accurate. Jhe above equation might be put 



^ . -In . ^ 2Txt 



?fi=-, ai cos Pi sin —^ < + Xi sin Pi cos ,., 



-TT 'lilt 



+ 0C3 COS ß. sin o -yp- t + ct;! sin ß^ cos '■) ,., 



+ &c. &c. 

 If we multiply this ])y 



sill — „, t dt or cos ,„ t dt, 



and integrate from < = 0, to <— 2', all the terms of the right-hand 

 member of the equation will in the usual way vanish, excepting one, 

 whicli is the term of the /th order. The equation is then reduced to 





. -lin , T 



ii„sin ,., t at ^ ^^ ocicos ß^ or 



r 



)(„ COS >_ t at ^ -^ «i sill ßi 



The first members of these eijuafioijs are simply the areas included 

 between the i-axis and the <'urve. 



sin / 2i7T \ 

 y-'""cos{ T 



between the limits < = (), and / = '/'. X(nv from the values of «„ and 

 corresponding t, different values of ij corres[)on(bng to /- are to be 

 found, and the curve carefidly ih-awn on section-paper. V>y means 

 of a planimeter, the cpiadrature «an l)e easily effected. If 



T T 



-:j- a, cos /3j == At ; — oLi sin ßf, == Al 



