TRANSACTIONS OF SECTION A. 621 



9. On the Electrolysis of a Solution of Amnionic Sv2pliate. 

 By Professor McLeod, F.B.S. 



10. ComjJensation of Electrical Measuring Instruments for Tewjierature- 

 Errors. By J. Swinburne. 



11. 4 Musical Slide Rule. By J. Swinburne. 



This is an instrument in which the distances between the marks are propor- 

 tional to the intervals — that is to say, the distances are proportional to the loga- 

 rithms of the vibration frequencies. This arrangement appeals to the eye and 

 gives clear ideas of the musical scale. In the accompanying scale the first fixed 

 scale is the ordinary equal temperament, the octave being divided into twelve equal 

 parts. The other fixed scale is the natural, and by means of the cursor the dis- 

 crepancies can be seen at once. On one side of the moving slide are two scales, 

 one being the natural and the other being the octave, divided into 53 equal parts, 

 commonly known as Bosanquet's cycle. By sliifting the slide the various intervals 

 can be added or subtracted. Thus, by putting c against p it is seen that d comes 

 opposite G, showing that the interval c-D is equal to p-g, The model being gra- 

 duated by hand is not correct throughout, but is sufficiently accurate to show the 

 working of the instrument. The cursor shows how nearly the cycle of 53 corre- 

 sponds with the natural scale. On the otber side of the movable slide are scales of 

 vibrations and logarithms. To find the vibration frequency of, say, E French pitch, 

 equal temperament, the line 5 is set opposite the mark on the equal temperament 

 scale, and the number of vibrations read ofl". For the natural scale the other mark 

 must be taken, as the scales are drawn so that the c's correspond ; the a's, there- 

 fore, do not come opposite. The logarithm scale is used for finding the logarithm 

 of any interval. 



It is suggested that such an instrument as this would give musical students a 

 much clearer idea of the nature of intervals and of the problems of temperament. 



12. On a certain Metliod in the Theory of Functional Equations. 

 By Professor Ernst Schroder. 



In order to prove that a functional equation does not follow from another given 

 one, it is indispensable to discover such a function as will satisfj' the latter equa- 

 tion without, however, satisfying the former. 



If, for brevity's sake, a function/ {x, y* .if two variables — supposed to be deter- 

 minatively invertible — is here denoted Hy ay, its inverse functions accordingly 



being represented by - and .r : y, then, for instance, from the equation 



Cu) ah = a : b 

 evidently will follow : 



Coo) {ah)c ={a:b):c. 



The impossibility, however, of deducing C^ from C„q may be demonstrated (and 

 it cannot be done in any simpler way) by means of the following table : — 



1=3,69,187245 | 4 = 6,93,421578 I 7 = 9,-36,7548 12 



2 = 1,47,298.350 l 5 = 4,71,532689 8 = 7,14,865923 



3 = 2,58,379104 | 6 = 5,82,613497 | 9=8,25,946731 



which in fact defines, within a system of nine numbers only, a function xy, so as 

 throughout to fulfil the equation Or.o, but not 0„. 



The meaning of the table is easily explained through the statement that its first 

 line is only an abbreviation for 1 = 33 = 69 = 90 = 18 = 87 = 72 = 24 = 45 = 51, where 

 S3 stands for/ (3, 3), and so on. 



