of Sulphuric-Acid Solutions. 141 



experiments of such accuracy, for we do not know whether 

 the present equations would still be applicable without modi- 

 fication. 



As to the uncertain break at 58 per cent, which Prof. 

 Pucker's equation successfully bridges over, I may point out 

 that even here my failure to recognize the sensible continuity 

 of the figure cannot be attributed to any fault in the method, 

 but to one in the operator. I invariably used the lath bent 



into the simplest form, y^- ^ , whereas, by apply- 

 ing the forces in a different manner it can be bent into the 

 wavy form, ; H r ~^^__L ir ? and when bent into this latter 



form I now find that it will fit over the supposed change at 

 58 per cent. ; but it may be remarked that if we adopt this 

 interpretation of this portion of the figure we shall not get a 

 straight line on differentiation ; and as the bulk of evidence 

 goes to show that the second differential is sensibly rectilineal 

 throughout the figure (a fact to some extent recognized by 

 Prof. Riicker, pp. 306, 307), it is probable that this one 

 portion would be of a similar nature, and that the interpreta- 

 tion of it as a wavy curve would be erroneous — a view which 

 seems all the more probable from the indications of changes 

 at this point in the case of other properties at this same 

 relatively high temperature, and from the existence of a 

 break in the freezing-points at low temperatures, which I 

 think no one would venture to question, even if the corre- 

 sponding hydrate had not actually been isolated. The present 

 case of the densities is the only one I believe in which a 

 wavy curve might be substituted for two of my simpler curves. 

 A wavy parabola can be found to bridge over the break at 

 58 per cent, as well as a wavy bent-lath curve, and in this 

 way the whole portion of the figure examined by Prof. 

 Pucker may be represented by two parabolas, and the use of 

 just as many constants as are required by Prof. Pucker's 

 equation ; the errors according to the two drawings are, 

 moreover, nearly equal ( (3 — a in Table I. are the errors 

 according to Prof. Rucker's equation, 7 — a the errors accord- 

 ing to the parabolas*), so that the magnitude of the error 



* The parabolas are y = -011767+ -00003782.1' - -OOOOloS&r'-, and 

 ij = -011767 + -00002917^ - -00000442^ - -0000000789.r ! , x = p- 73, 

 The constants have been deduced from the readings of a curve instead of 

 directly from the experimental values, and the average error is, conse- 

 quently, rather larger than it would otherwise have been. It would in 

 any case be somewhat larger than that according to my original drawing. 

 for the equations extend a short distance beyond the points at which I 

 believe breaks occur, namely, 51 and 78 per cent. 



