308 TRANSACTIONS OF SECTION A. 
my respectively we call the orders of coincidence of R(e) with the p-adic factors of 
the fundamental equation f(x) = 0. 
The coefficient of «"-1 in the number R(e) we call its principal coefficient. The 
orders of coincidence of such number relative to the prime p will be integral multiples 
of certain numbers 1/y,,..., 1/y,, where »,,.. ., v, are factors of n,, . . ., n, 
respectively. We consider the aggregate of numbers R(e) possessing an assigned 
set of orders of coincidence relative to p. The necessary and sufficient condition 
that the principal coefficient in the aggregate be integral is that the assigned orders 
of coincidence have a certain set of values. This particular set of orders of co- 
incidence defines adjointness relative to the prime p with regard to the fundamental 
equation. 
i If we start out from a sufficiently general rational form R(e) with its coefficients 
represented in p-adic form and impose on it the conditions requisite in order that it 
may possess a certain set of orders of coincidence relative to the prime p, these con- 
ditions take the form of a succession of independent congruences relative to the 
prime p imposed on the coefficients of the powers of p in the p-adic coefficients of 
the powers of «. We find a formula for the number of these conditions. We also 
assign sets of orders of coincidence 7,”),... za corresponding to all primes 
p, these orders of coincidence being 0 with the exception of a finite number among 
them. Such a system of orders of coincidence we call a basis of coincidences. We 
define complementary adjoint bases of coincidences and derive the analogue of the 
complementary theorem in the theory of the algebraic functions. 
3. The Green’s Function for the Equation 7?u + k?u = 0. 
By Professor H. 8. Carsnaw. 
4. The Evolute of the Limacon. 
By Professor W. H. H. Hupson, M.A., LL.M. 
The equation of the Limagon is taken in the form r=a (1+ cos 6), a will be 
made 1, and the abbreviations used 1—e? =f, 1—4e?=k, e(1+e) /(1-++2c) =e, 
e(1—e) / (l1—2e)=c’. 
e=-3 €=3 
D 
D \ 
ee 
Cc Cc 
C co Cc 
D’ os 
D 
The Limacon is symmetrical about the x- axis; so therefore is its evolute. 
(1) When e=0, the Limacon is a circle ; the evolute is a point, the centre of the 
circle. 
(2) When 4 >e>0,theevolute is a closed curve with four cusps pointing outwards. 
These cusps are D,(ef, e?/f), D',(e f,-e Wf), C,(c,0), C’, (c’,0). _ All four lie on the 
circle y?+(a—c) (vc—c')=0. Ase increases, this at first star-like figure grows larger, 
the total height approaches 4/3/4 as e approaches 4, the breadth (parallel to the 
x- axis) increases indefinitely. ; 
(3) When e=}, c’ becomes 0, the height becomes /3/4. The cusp C’ is at 
