TRANSACTIONS OF SECTION A. 453 



and also for its uses in cases such as 



or 2 , ar 3 , x-*, or 213 , &c.,&c; 



in any of which x may denote any numeric whole or fractional. 



Such expressions as the square of the unit of time are not to he approved of. 

 That expression, for instance, is too like such comhinations of words as a square 

 second, or a square minute. It is not really quite so unreasonable, however, because 

 there is a good meaning sometimes intended though badly expressed, when the 

 square of the unit of time is spoken of, that unit being then regarded as a variable, 

 and the' true meaning being usually just what may be distinctly stated as the second 

 power of the change-ratio of the unit of time. A second is essentially a constant 

 quantity of time ; and the square of a second, a second squared, and the second power 

 of a second of time, are all of them essentially meaningless conjunctions of words. 



It is to be observed that anv ratio is a numeric, or may be treated as such to 

 any degree of approximation we please ; and so we can have the second power or 

 any other power of a change ratio ; and we are entitled properly to write as a frac- 

 tion any power of any change-ratio divided by any power of the same or of any 

 other change-ratio. That fraction will be itself a numeric, and may properly form 

 one side of an equation having a numeric for its other side. 



It follows, then, under the views already offered, as to legitimate and illegiti-. 

 mate modes of expression and notation, that such notations as 



, yard _ 1 foot 



(minute) 2 ~ 1200 (second) 2 ' 



which is given in Dr. Everett's Treatise (page 4), as a very expressive notation now- 

 becoming common, are not commendable, and are such as it is desirable promptly 

 to reject. We might quite rightly note that 



1 yard_ 1 /minute\ 2 

 foot ~ 1200 V second/ 



but it would be illegitimate to pass from this, by imitation of a real algebraic pro- 

 cess, so as to write 



, yard 1 (minute) 2 

 foot " 1200 (second) 2 ' 



and thence further to make a pseudo-multiplication of both sides by foot, and a 

 pseudo-division of both sides by (minute) 2 , and so to bring out the seeming equa- 

 tion above objected to. 



The name dimensions of units is subject to a distressing ambiguity. It might : 

 mean the greatness or smallness of them ; and indeed a dimensional equation is for 

 the very purpose of telling how the greatness of some units changes in accordance- 

 with changes made in the greatness of other units. This is not, however, the idea 

 which is attached to the word dimensions in dimensional equations. It is mentally 

 associated rather with such notions as the three dimensions in space, length, 

 breadth, and thickness (not to say also with the fanciful notions, so often now put 

 forward, of a fourth dimension in space, or a 2|th, or 4§th, or an infinite number of 

 other alleged dimensions in a dreamland space, not found in our world or conceived 

 in our brain). It has, in fact, to do with change-ratios, or powers of change-ratios 

 of quantitative units, whereby the magnitudes of the various units are mutually 

 connected, and some of them specified by reference to others. There is, I may re- 

 mark, for instance, in Dr. Everett's book, one article on Dimensions of Units, and 

 another on Dimensions of the Earth. Now the word dimensions in these two cases 

 has totally different meanings. 



10. On Lead and Platinised Lead as a Substitute for Carbon and Platinised 

 Silver, in Leclanche, Bichromate, and Smee's Batteries. By Edward 

 T. Hardman, F.C.S., 8fc. 



The chief objections to carbon as a positive plate are: (1) the difficulty of 

 preparation ; (2) the limited size of the plates ; (3) their brittleness ; and (4) th« 



