TWENTY-THIBD ANNUAL MEETING. 65 



The most remarkable find in this region, however, was made on the Whitewater 

 river, twenty miles east of Wichita, and seems to date back in geologic times prior 

 to the last general submergence of the country. At a point near Augusta the river 

 has gradually deflected to the east cutting its way slowly into the valley, exposing 

 the strata of a perpendicular bank some forty feet in height. The top of the bank 

 is the general level of the valley, in which was growing an oak tree five feet in 

 diameter. The first eighteen inches in depth is the usual dark surface soil ; the 

 next eight feet is yellow clay, apparently of the Loess formation common in Kansas. 

 This clay rests upon a former surface soil two feet thick, of rich black loam ; the 

 line between the two is sharply defined. The black soil merges into clay below, 

 which extends down to gravel resting upon the bed-rock of the river. On the sur- 

 face of the black soil and under the eight feet of clay I found the remains of a 

 camp, containing broken pottery, charcoal, ashes, burnt bones, and stones such as 

 would be used in a camp fireplace. The bones resembled the legbones of deer. 

 The black stratum of soil was undoubtedly the surface of the valley, rich in vegeta- 

 ble and animal life, at the time the aboriginal, antediluvian people feasted around 

 their camp-fire, and broke their soup-bowl. 



We are evidently not the first settlers of Kansas. 



DIFFERENTIALS OF THE SECOND AND HIGHER ORDERS. 



BY E. MILLEE, LAWBENGE. 



This memoir is written as an answer to one of many questions that have been 

 addressed to the Department of Mathematics of the University. Such inquiries 

 are of almost weekly occurrence. They cover ground that extends all the way from 

 Problems in Percentage to "Curves of Pursuit." They embrace the -'settlement of 

 estates;" disputes arising from the "foreclosure of mortgages;" the winding-up of 

 "joint-stoek companies;" the contents of cisterns; the "horse power" of mill- 

 dams; the properties of various kinds of curves; the Theory of Probabilities, and 

 the elimination of differentials. More than once has the chair of mathematics been 

 called upon to elaborate the principles, the notati(jn, and the application of differ- 

 entials, and to show why all differential expressions, of second or higher orders, 

 when compared with those of the first order or degree, vanish completely from the 

 work in hand, without affecting the result. To have answered this question in 

 detail, by unfolding the demonstration step by step, as generally given by either of 

 the methods of the Calculus, would have required more time and leisure than I had 

 at my command. The proposer of the question was evidently laboring under the 

 impression that to throw away the quantity dxdy from such an expression as 

 a;dy-\-ydx-}-dxdy, was an absurd thing to do; believing that if dx and dy each had a 

 value, however small, then dxdy would also have a value, although indefinitely 

 smaller than the former. Now, whether we use the Infinitesimal Method, in which 

 "a quantity is conceived under such a form, or law, as to be necessarily less than 

 any assignable quantity," and according to which infinitesimals of the second, ffiird, 

 and higher orders, may be dropped as not affecting the result; or, the "Method of 

 Limits," as enunciated by one of the discoverers of the Calculus, that "quantities, 

 and ratios of quantities, which in any finite time converge continually to equality, 

 and before the end of that time approach nearer the one to the other than by any 

 difference, become ultimately equal," ive know that dropping the higher orders will 

 give beyond a peradventure the exact result. 



