78 ROYAL SOCIETY OF CANADA 
bers found by multiplying D° by any combination of the v primes divid- 
ing e is a number whose greatest square divisor is D?, and which is itself 
a divisor of m. We thus make 2”, divisors of m, correspond to D’, But 
in this way we must get every divisor of m, since every divisor of m has 
a greatest square divisor. 
If we denote the divisors of m of the forms 4 À + 1,444 3 by M 
and N we find = 2” = M+ N. 
D*:m 
Next, the sum = 2”, where ifm = de, e contains a prime of the 
ad’: m 
form 4 À + 3, which is the second part of the difference, is equal to 2 N. 
For to any such divisor d we can make correspond 2’—, divisors of m. 
whose greatest square divisor is d*, and which are of the form 4 À + 3, 
To do this we multiply d* by any combination of the 7 primes dividing 
e, which contains an odd number of primes of the form 4 À + 3. 
Let there be w of these primes in all Then we must pick out 
an odd number of them which will give a product of the form 4A + 3 
which, when multiplied by d? (of form 4 À + 1) and by any number of 
primes of form 4 À + 1, gives finally a product of the form 4 À + 3. 
But the number of ways of picking an odd number of primes from 
jis equal to the number of ways of picking an even number, as appears 
from the expansion of (1—1”)= 0. Hence the 2” numbers whose 
—1 
greatest square divisor is d° fall into two lots, each lot containing ar 
numbers, and being of the forms 4 À + 1, 4 À + 3 respectively. 
Hence 3 2”— 252” =2 NV, since, as before, every divisor of m has 
asm ' asm 
a greatest square divisor. 
Hence 4022" 1 So prea ye) 
DP Em 

=4(u— VN) 
This may be‘written in _thejform 4 2 (—1) IT here d is every 
d:m i 
divisor of m. 
Special Biquadratic Involutions and the Transformation of Elliptic 
Integrals, 

Let re ee be an elliptic integral in homogeneous form, f being a 
biquadratic form. We use the notation for invariants of Clebsch, Binäre, 
Formen. Then Hermite has shown that if we put y,= — A, y¥,=2f, 
and use the relation 2 T°? = — H? +H? — fs, and also put g, = $7, | 
