MATHEMATICAL SECTION. 103 
In the theory of elliptic functions, sin amu, cos amu, A amu 
(Jacobi’s notation) or snu, cn, dnu (Gudermann’s notation), the 
elliptic quadrant K (Jacobi) is the numerical unit of their period. 
Consistency requires the use of the quadrant as a unit for trig- 
onometric functions also. Let _| denote a circular quadrant (ordi- 
narily denoted >) ; then we have, by the notation just explained, 
ma | 
inet NMEA ip 3 
—_S_= ose ch (= K of Jacobi). ( ) 
oO 
The complementary integral then 
a 
d 7 ° 
STFS =! © K of Incobi. (4) 
If n is an integer, then, and only then, (n_|)y =n _|y. (5) 
Thus we should be careful in distinguishing between integrals 
such as 
2 _] a 
? 
i Pe 9 aA ean 
Oy \ rapa tb Se 
According to the system of notation just explained, it is unneces- 
sary to use the Jacobian am or the Gudermannian n, neither of 
which define the functional relation completely, and we write simply 
sin g = sin u_y (= sin amu of Jacobi or 
snu of Gudermann) 
cos = cos u_y (= cos amu of Jacobi or 
enu of Gudermann) 
V1i—7sin’g= Ag= Au_y(= A amu of Jacobi or 
dnu of Gudermann) (6) 
I remark that none of the usual notations indicate the modulus, 
and a grave objection to Gudermann’s is that it is apt to give the 
impression that snw and cnw are not an ordinary sine and cosine. 
I shall now give in this notation a number of well-known relations, 
of which use will be made hereafter. The theorem of addition is, 
if w and v are two integrals to the modulus +, 
