TRANSACTIONS OF SECTION A. 325 



amount. This causes the temperature of the vapour-space and oil surface to 

 rise less rapidly in the Abel-Pensky and a reasonable explanation is thus 

 afforded of the differences between the two types. In addition, direct' com- 

 parisons of the three types of apparatus were made with a series of oils of 

 various flash-points. A practically constant systematic difference of slightly 

 over 1° F. was found to exist between the Abel apparatus and the Colonial 

 type of Abel-Pensky and a difference of nearly 4° F. between the Abel and the 

 German Abel-Pensky. 



The conclusions given in this abstract are based upon the results of about 

 fifteen hundred experiments. 



MONDAY, SEPTEMBER 4. 



Joint Discussion with Section G on Aeronautics. Opened by 

 A. E. Berriman. — See page 481. 



Department op Mathematics. 



The following Papers were read : — 



1. Proofs of certain Theorems relating to Adjoint Orders of Coincidence. 

 By Professor J. C. Fields. 



In this Paper the author starts from an arbitrary algebraic equation. 



F(r, u) = (M-P,) . . . (a-P.) = (1) 



where P, . . P„ are power-series in an element z — a (or -) with exponents 



integral or fractional. On adding / (r, u) to the reduced form of any rational 

 function of (r, u), such function can be represented in the form 



(« - Q,) ...(«- Q„) (2) 



If the orders of coincidence of the rational function with the n branches of the 

 equation corresponding to the^ value r = a (or r = oo)each exceed by 1 the orders of 

 coincidence requisite to adjointness, on properly co-ordinating the series P and 

 Q it is readily shown that we can write 



or ( , 



Q r = P, + /1\«-S, ( r = 1 n 



where the exponents <V are > and where the series S r are integral with regard 



to the element r — a(oi~\ On substituting these forms for the series Qr in the 



product (2) it is readily seen that the coefficient of u"- 1 in the reduced form 

 of the rational function represented by the product must be divisible by the 



element r — a(or~Y Where it is the value r — a which is in question it is easily 



6hown that the degree in (r, u) of the reduced form is < N — 1 where N is the 

 degree of / (r, u). In the case of a finite value r = a it furthermore follows that 

 each coefficient in the reduced form is divisible by the element r — a if the equa- 

 tion (1) is integral with regard to this element. The statements just made are 

 evidently equivalent to the following theorems : — 



(1) In the reduced form of a rational function of (r, u), which is adjoint 

 for the value r = o(or r = oo), the coefficient of u"- 1 is integral with regard to the 



element r — a (oi _ V 



