464 



SCIENCE 



[N. S. Vol. LI. No. 1323 



tin or glass receptacle quickly dissipates it; 

 once lost, it can not be restored. Observers 

 have been able to detect the sound from a New 

 England beach sand over 400 feet away, when 

 a small bagful is suddenly shaken. 



While the analogy to the snow crystals may 

 account for part of the phenomenon in some 

 cases, it can not account for the singing of 

 limestone, coral or other non-crystalline sands. 

 Moreover, when one walks barefooted on 

 musical sands, or runs the hand through them, 

 there is felt a distinct tingling sensation. To 

 some, this has suggested an electrical prop- 

 erty. • The latest and most plausible theory is 

 that upon clean, dry sands, atmospheric gases 

 condense, iust as gases will adhere to particles 

 of some metallic minerals and not others, and 

 that the sounds and the sensations described 

 are due to the disturbance of these air cush- 

 ions. At any rate, the sensation experienced 

 when walking barefoot through a patch of 

 musical sand is very similar to that felt when 

 the hand is immersed in a solution in which 

 nascent oxygen is being generated.^' 



By the way, I wonder if it has ever occurred 

 to any archeologist that a possible explanation 

 of the " Vocal Memnon " which Strabo and 

 other travelers attested some two thousand 

 j'ears ago, might be the presence near the 

 colossi, of musical sands, long since buried 

 by the drift from the Libyan Desert. 



Albert R. Ledoux 



modern interpretations of 

 differentials 



To THE Editor of Science : Professor E. V. 

 Huntington, in an article entitled " Modern 

 Interpretation of Differentials" (Science, 

 March 26), states with reference to the defi- 

 nition lim Ay = 0, lim NAy = dy, that, " The 

 inevitable consequence of such a definition is 

 that dy = 0, which is futile." Every school 

 boy in the theory of limits knows that this is 

 not true when N varies. 



To take his figure of a gi-aph of a function 

 y = f(x), it is logically correct to denote a 

 point on the graph by P{x, y) without sub- 

 scripts, and P'(x-\-Ax, y-\-Ay) is any other 

 point on the graph, where PQ = Ax, QP' = Ay. 



Produce PQ to PB' = NAx = A'xj and draw 

 R'S' = NAy = A'y, parallel to OY. Then 

 S'(x -ir A'x, y-\-A'y) is any point on the pro- 

 duced chord PP" (i. e., variation in the same 

 ratio is along the chord). 



AS' 



Professor Huntington asserts that S'(x + 

 A'x, y -\- A'y) inevitably approaches coinci- 

 dence with P{x, y) when Ax, Ay, approach 

 zero, although it is obvious that it may, if N 

 increase appropriately, approach any chosen 

 point S(x-\-dx, y-\-dy) on the tangent at 

 P{x, y), so that lim A'x^dx, lim A'y^dy. 

 Variation in the first ratio is therefore upon 

 the tangent. 



Professor Huntington should also have in- 

 vestigated the historical questions involved 

 before venturing to assert that the above 

 theory of differentials " would prove highly 

 misleading to the modern student." It is a 

 sad commentary on the present state of the 

 calculus in respect to its fundamental ideas, 

 when we note the variety of explanations of 

 these ideas by authors with little historical 

 knowledge, all of whom, no doubt, would term 

 their productions " modern," though most ex- 

 planations will be found to date back several 

 centuries, if they be anything more than 

 vaporizing. 



Sir William Rowan Hamilton in his Ele- 

 ments of Quaternions (Bk. III., p. 392) states 

 that ordinary definitions by derivative meth- 

 ods do not apply in quaternions, and that 

 after a careful examination of the Principia, 

 he would formulate and adopt Newton's defi- 

 nition as follows: 



