286 FORMULZ FOR THE COSINES AND SINES 
entirely arbitrary, but may be shown to be subject to the condition 
that 77’ asind is constant, where 7 7’, are the two forces, @ is 
the shortest distance between their lines of action, ¢@ the angle be- 
tween these lines. (Cambridge 8S. H. 1833.) For let the couple 
be changed so that its forces are 7,7, and a is its arm, and let it be 
placed so that 7 acts at the same point as R, and the arm is at right 
angles to R. Then T and & being compounded into 7’, the angle 
between 7’ and 7 will be ¢, and we have J’ sind = FR cosd. Also 
G cos@ being constant, and Ta being equal to G, Ta cos@ is constant, 
and therefore, since # is constant, we have 7'7'a sin¢ also constant. 
This can also be expressed geometrically by saying that if the two 
forces be represented in position and magnitude by two straight lines, 
and the extremities of these lines be made the angular points of a 
pyramid, the volume of this pyramid will remain the same, whatever 
way of reduction be chosen. ‘This elegant proposition was first 
given (so far as I am aware) in the Ladies’ Diary, 1836. 
In a subsequent note the analogous propositions in the motion of 
a rigid system will be discussed. 
FORMULA FOR THE COSINES AND SINES OF 
MULTIPLE ARCS. 
BY THE REV. GEORGE PAXTON YOUNG, 
KNOX COLLEGE, TORONTO. 
§1. Take the expressions, 
Do 2) Tle Py, Tes els -cek sete cee cece 
So that, ¢ being any quantity, and c a number greater cans zero, the 
relation 
(one =e wd sseyicide oases ae eee pee 
always subsists. Hence T,=1—22?, &c. In like manner, take the 
expressions, 9 
£=0, 2,1) t55¢55 &Cis.5. sets ds cid tc ICO 
So that, ¢ being any quantity, and c a number greater than zero, the 
the relation 
te+, = t—t?t,_, Maral aivfehnolelete'aisinavateleleysietetsie (4) 
