538 KEPOBT— 1886. 



There are pairs of ■wires of each of the metals platinum, gold, palladium. 

 These are hung side by side from the sa7ne top support, and one of the wires in each 

 pair carries a light load (^-^ of the breaking weight), the other a heavy load, about 

 f of the breaking weight. A comparison is made between the pulling out of the 

 heavily loaded wire and the lightly loaded wire by means of apparatus described in 

 former Reports. 



The wires were set up on May 3, 1879. By May 6 they had all come to a 

 fairly steady condition, though the heavily weighted wires were running down to 

 an extent just perceptible from day to day. 



From May 6, 1879, till August 7, 1880, the running down of the heavily 

 loaded wire in comparison with the lightly loaded wu-e was in the case of platinum 

 1'15 mm., on a length of 1553'33 centimetres ; in the case of the gold wire 

 1*45 mm, on 1552-98 centimetres. 



From August 7, 1880, till March 3, 1886, a further running down has taken place, 

 which amounts in the case of the platinum to 0'40 mm., and in the case of the gold 

 to 0-80 mm. 



Mathematical Sub-Section. 



1. Report of the Committee for Gahulating Tables of the Fundamental 

 Invariants of Algebraic Forms. 



2. On the Rule for Contracting the Process of Finding the Square Root of a 

 Number. By Professor M. J. M. HiLli. 



The rule is this : — 



See ' Todhunter's Algebra,' Art. 246. 



When n+\ figures of a square root have been obtained by the ordinary method, 

 n more may be obtained by division only, supposing 2/1 + 1 to be the whole 

 number. 



After giving the demonstration Todhunter says : ' The above demonstration im- 

 plies that N (the number whose square is to be found) is an integer with an exact 

 square root ; but we may easily extend the result to other cases.' 



He does not, however, give a general proof of his statement. 



The object of this paper is to show that the result of the division may exceed 

 by unity the remaining n figures of the square root, and then the rule fails. 



The proof given in text-books on algebra is applicable to the case when the 

 square root can be exactly obtained as an integer, but not to all other cases. 



On the Explicit Form of the Complete Cubic Differential Resolvent. B]f 

 the Rev. R. Haelet, F.B.S.—See Reports, p. 439. 



4. On a Geometrical Transformation. By Professor R. W. Genese, M.A^ 



Let SP, ST' be parallel radii of two circles in opposite directions, then PP' 

 Fig. 1. passes through a fixed point O on the line 



of centres SS' (viz., O is the internal 

 centre of similitude). Let PP' meet the 

 circles again at QQ'. Then the locus of 

 the intersection of SQ, S'P' is a conic of 

 foci S, S'. For in fig. 1 SP = SQ .-. RP' 

 = RQ .-. S'R -SR = S'P' + SQ = constant 

 (if the circles cut, we find S'R + SR 

 constant). 



Now let the circle round S' become a- 

 straight line, S' passing to infinity. Then any point on SL (the perpendicular to> 



