160 REPORT— 1898. 



following useful property possessed by the homologues. (We may omit 

 the adjective when no ambiguity is likely to arise.) 



' If for two curves (1) and (2) 3;.2=^'^i ^^id 2/2=^^1) then a;J=a;} + log a 

 and 2/2=2/1 + ^^a ^ > 01' ^h® logarithmic homologues will all be similar 

 curves, but differentlyplaced with regard to the axes, such that the one curve 

 may be brought into coincidence with the other by a shift of which the 

 coordinates are log a and log b.' ^ 



It should be noticed that this shift of the homologue is one of pure 

 translation. 



The graphic method of homologues was again used by Professor 

 Reynolds in discussing experiments on the flow of water. - 



2. Mr. Human has since patented the manufacture of sheets of paper 

 ruled logarithmically.^ A short account of the use of logarithmic 

 coordinates may be found in GreenhiU's ' Differential and Integral 

 Calculus,' 1896 edition; directions for the use of the ruled sheets, with a 

 number of easy examples, are supplied by the publishers of the ruled 

 paper, and the valuable aid to computation which this method gives may 

 now be considered well known. 



3. The power of readily moving homologues on the logarithmic paper 

 to represent changes in the original equation was greatly facilitated by 

 the invention of scale lines by Mr. Boys.'* To explain the use and method 

 of construction of scale lines Mr. Boys drew a chart of wave and ripple 

 velocities, which by its wonderful generality was calculated to emphasise 

 the power of the new method of discussing curves by means of their 

 homologues. 



Definition and an Example of a ' Translatant.' 



4. Let us consider the equation 



*> — ^ + ) 



2t \p ' 



which is the equation that Mr. Boys took to illustrate the method. It is 

 of the form 



v^ = a\+-. 

 \ 



Let a, b, v, and X become a', b', v \/ —r ^"^^ ^ A / ~7 " ^^® equation 



remains unaltered ; thus, if g, r, and p all change, it is possible, by merely 

 shifting the homologue, to represent the new equation. The homologue 

 of this equation possesses the property of translation. 



5. It is of great service in any subject to have a definite nomenclature, 

 and a cihrve whose homologue possesses this property with respect to any 

 particular quantity might be called a ' translatant ' with respect to that 

 quantity. The term translatant can also be applied without risk of 

 confusion to the homologue. 



6. Definition. — A curve or its homologue is a translatant with respect 

 to any quantity when an arbitrary change in that quantity is equivalent 



' PUl. Irani., 1879, Part II., p. 753. 



"^ Ibid., 1883, Part III., p. 947. 



' Published by Beaves and Stephenson, 8 Princes Street, Westminster. 



< Nature, July 18, 1895, p. 272. 



