30 Observation^ on the Notations employed in 



the following equation is said to express the connexion be- 



dz . dz 

 tween -=— and —. — , 

 dx ay 



dz dz dz dy . 



dx dx dy ' dx ** 



from which we should naturally conclude that 



dz dy 



= . . 



dy d x 



'* Now we ask, what connexion does this establish between 



-j — and — j— ? Certainly none. In order, however, to ex- 



plain how equation (1.) does represent such a connexion, we 



are told that — j— - on the left-hand side of the equation does 



d z 

 not mean the same as -= — on the right-hand side;' an ex- 

 d x 



planation not likely to be very satisfactory to a learner." 



The above are the strongest reasons, indeed the only rea- 

 sons, that I have seen advanced for adopting the new notation. 

 I have made these extracts, in order to set the writer's most 

 cogent arguments before the reader ; still, I would advise him 

 to peruse the book and form his own judgement. 



With regard to the above quotations, I wish briefly to re- 

 mark, that it seems to me the reference to Lacroix is not suf- 

 ficiently explicit to do justice to that work. It should be ob- 

 served that the two differential equations taken from that work 

 belong to two sections of the curve surface, the equation of 

 which is u = 0; and " the dz of the first equation must not 

 be confounded with that of the second, for they are both only 

 'partial differentials, as has been remarked in No. 120." 



I do not pretend to determine the point, but I am impressed 

 with the notion that a careful perusal of Arts. 120 and 127 

 in Lacroix, upon which the equations cited depend, will clear 

 up the mystic appearance which they bear in the pamphlet. 



With respect to the equations d z = —r- .dx and dz = -r .dy, 



CI 3C C'T/ 



said to be given in Lacroix by way of explaining the mystery, 



this is what Lacroix really does say : — 



" When we have dz = pdx, dz is the differential of the 



ordinate of the section parallel to the plane of x and z : and 



similarly, dz = qdy is that of the ordinate of the section 



d z d z 



parallel to the plane of y and z;" here p = -^- and q = -j- : 



I am unable to perceive that these equations were intended 



