396 Professor Hamitron on Conjugate Functions, 
secondary. In like manner, we can conceive sums of more than two step-couples, 
and may establish, for such sums, relations analogous to those marked (5.) and (6.) ; 
thus, 
(ey, %)+(by, bay +(ay, as)—Cer+ bit a1, oot byt a4), (s 
= (an, ay) +(d,, db») +(e, 20) &e. t ) 
We may also consider the decomposition of a step-couple, or the subtraction of one 
such step-couple from another, and may write, 
(b,, bo) —( ay, ay)=(b,— a,, by— a4), (9.) 
(b,, b,)—(a,, a.) being that sought step-couple which must be compounded with or 
added to the given component step-couple (;, 4,) in order to produce the given 
compound step-couple (»,, b»). And if we agree to suppress the symbol of a null 
step-couple, when it is combined with others or others with it in the way of addition 
or subtraction, we may write 
(41 a,)=(0, 0) +( a1 ay)= +(ay ao), 
(-— 1, —a,)=(0, 0)—( a1, ayy= —(a,, ay), Os) 
employing a notation analogous to that explained for single steps in the 35th article 
of the Preliminary Essay. Thus +(,, 2.) is another way of denoting the step- 
couple (@,, 2.) itself; and —(,, ®,) isa way of denoting the opposite step-couple 
(— 41, — #2). 
On the Multiplication of a Step- Couple by a Number. 
3. From any proposed moment-couple (a;, 42), and any proposed step-couple 
(a, a), we may generate a series of other moment-couples 
So (£1, E>»), (4), Ey); (Ai Aa); (Bi, Bp); (Bi, B’2) eee (11.) 
by repeatedly applying this step-couple (a), a), itself, and the opposite step-couple 
—(a, ), or (—a,, — a), ina way analogous to the process of the 13th article of 
the Preliminary Essay, as follows : 
