402 Professor Hamitton on Conjugate Functions, 
tion (38.), when a, and a, are not both null steps, it is necessary and sufficient that | 
the factor 
a, (a, + Ya ay) a ay =(a, + by ay)? — (yn ots 4y,*) a” (4.4.) 
should never become null, when a, and a, are not both null numbers; and this con- 
dition will be satisfied if we so choose the constants of multiplication y, y, as to make 
yn t+iy’<0, (45.) 
but not otherwise. Whatever constants y, y2 we choose, we have, by the foregoing 
principles, 
(ec, 0) (0, ¢) (O, ¢) 
Se lO) 2. = (Oeil) ae (0) ia (46, 
AO) » 93 (ec, 0) (O, 1); (ORS 20); ) 
and finally 
25.0) 8( sae, »)., (47.) 
(O, c) yi ¥y ie) 
because, when we make, in (43.), 
¢,=0, a=1, B,=1, B’=0, (48.) 
we find 
Gras et ; (49.) 
yi yi 
so that although the ratio of the pure primary step-couple (c, 0) to the pure second- 
ary step-couple (0, ¢) can never be expressed as a pure primary number-couple, it 
may be expressed as a pure secondary number-couple, namely (0, aed if we choose 
O, as in (36.), for the value’ of the secondary constant y,, but not otherwise : this 
choice y,=0 is therefore required by simplicity. And since by the condition (45.), 
the primary constant y, must be contrapositive, the simplest way of determining it’is 
to make it contrapositive one, y,= —1, as announced in (36.). We have there- 
fore justified that selection (36.) of the two constants of multiplication ; and find, 
with that selection, 
Ce, ee 50 
(20) (0, -1), (50.) 
and generally, for the ratio of any one step-couple to any other, the formula 
( b,, b.)_ (Bi c, Pr c) = (Biss =F By ay ‘ B, a —B, ay J (51.) 
CA 2) Care ape) 
2 2 
ay +a, a; +a; 
