TRANSACTIONS OF THE SECTIONS. 
5 
On the Summatim of certain Circular Functions, 
By the Rev. Professor Jarrett, M.A. 
l!itsquatioD-=8in«—-.8m2®+-^.8m3®— &c. is found iamany works on 
2 2 3 
fcdiferential calcalus, and various paradoxical conclusions have been deduced froni 
i. Hie design of this paper ia to show that the equation is not true ns R generm 
fcom, however correct it may be when the value of x is confined within certam 
Uu. 
If d* eqaatioa were generally true, then the series on the right side ought, when 
Bunded according to ascending powers of x, to have zero for every coefficient after 
Ikt of the 6rst power, ^ 
httisg for t terms of this scries, and substituting x — j ^ j 
hwj, we shall find the coefficient of (—1 )’’**'*-^—^ 
^thesR terms are alternately positive and negative, it is convenient to consider 
Bpwtely the case of an even and of an odd number of terms; and wc shall nave 
-j^.(128r8+256f?-224dH112/-’-170+&c. 
-^j.(32I«+144r®+240l<+l8(»3+fiO^H9^+l) 
+j^.(128(«+J68r?+1792<o+2016r®+1120d-*+3363f+112i=+33/+l)- &c. 
llwill he seen, therefore, that in the case of an even number of terms, however 
the first power of « vanishes, instead of being the only one that is lett; wmie, 
* the other hand, not one of the other odd powers disappears. Also, it an oou 
2 ^ of terms are taken, the first power of x does not indeed vanish, but its co- 
'«the least of the whole series, while the coefficient of every other term ne- 
and greater as the number of terms in the original series is mere 
JjKther. therefore, we stop at an even or at an odd number of terms, the equa- 
®flueation cannot be oCTfroWy true. , . «„„ai-lnn 
lamination of tlic usual methods of demonstrating the supposed equation 
*™'W that the correct result is 
* . 1 1 1 . 
sin X ——. sin 2#+-^. sinJr—&c. +—• 
fx C0B(f+l)J+ 
2[/ *=l jq-cos ^—l.siuJ? 
On a new Proof of the Principle of Virtual Velocities, 
By the Rev. B. Price, M,A. 
On (Ac Fundamental Laxos of Motion and Equilibnwn. 
^ By W. Spottiswoode, B.A. 
^^Pjper of an d ;inon deduction of the laws of motion 
of a fundnincntal connexion between the sciencca 
