50 SCHOOL SCIENCE AND MATHEMATICS

quantities of mineral matter to waters flowing through them; the

streams entering Lakes Michigan and Huron, on the other hand, traverseregions made up largely of soluble sedimentary rocks�limestones,

sandstones, and clays�which yield to the waters comparatively large

amounts of the calcium and magnesium compounds. The difference

in mineral content between Lake Huron and Lake Michigan results

from the dilution of the water of Lake Huron by the softer water of

Lake Superior. Lake Erie is highest in incrustants because it re-ceives out only the waters from Lakes Michigan and Huron, but thedrainage from immense areas of sedimentary rocks in Indiana andOhio and in the province of Ontario. The waters of Lake Ontarioare softer than those of Lake Erie because they have been diluted by

drainage from areas of crystalline rocks in New York and the Cana-dian province. The water of all the lakes is modified greatly bythe large amount of rain and snow directly added to their surfaces,serving to measurably lower the proportion of mineral matter. Thequality of the water in each lake varies but little from month tomonth; the variation is, however, measurable, and in winter themineral content is higher.The lake water is as a rule very clear, containing practically no

suspended matter except in times of storms, when the sand and de-posited organic matter near the shore are stirred up by wave action.

It has been computed that more than 29,000,000 tons of mineral mat-ter, chiefly calcium compounds, held invisibly in solution, are annuallycarried from Lake Erie to Lake Ontario over Niagara Falls; but,notwithstanding this enormous amount of dissolved material in "theirwaters, the investigation, proves conclusively that as long as theGreat Lakes remain unpolluted by sewage they will afford the regionsurrounding them the best available water supply for municipal andindustrial uses.�U. S. Geol. Survey Bul. 28^.

A SECOND NOTE ON THE SOLUTION OF THE EQUATIONa + - = c for ^c = y = o.

^ y

BY M. 0. TKIPP,

College of t7ie City of New YorJc.

In the November number of this journal, Professor Miller wrote indefense of one elementary algebra in not including (0, 0) as a solutionof one given equation. As to whether (0, 0) shall be considered asolution, depends upon the point of view an author takes. If he con-

siders a- or what is the same thing � as having a meaning, viz.;o o

infinity, then he cannot consistently exclude (0, 0) as a solution.Such quotations as the following from well-known algebras show the

prevailing attitude with regard to the meaning of A .o

ChrystaVs Algebra, Part I, page 322:

SOLUTION OF AN EQUATION 57

"P >Q will certainly == co if P 0, Q = O."

Fine’s College Algebra, page 226:

"We therefore assign to �, and in general a- when a ^ o the ’value’^ o <

oo writing � == oo .)’<9

Beman and Smithes Academic Algebra, page 133:

"The form a-. This form should he interpreted to mean an expres-o

sion whose absolute value is infinite."The meaning of multiplication and division by zero is carefully stated

in Well’s College Algebra, page US/note: "It must be clearly under-stood that no literal meaning can be attached to such results as

� ==<^i a; X <9 == o , or � = co ;CO ’ ’ o

for there can be no such thing as multiplication or division when themultiplier or divisor is 0 or oo."

Since these representative authors with a host of followers have

given a meaning to -1 , they must either change their texts or else,o

as the texts now stand, must accept the above solution if they wouldbe consistent.No tribunal of eminent authority has yet denied that it is sometimes

of advantage to assign a value to -~ . Certainly this is the case in the°s

above equation where it enables us to obtain a value of y correspondingto x == 0. and furthermore makes it possible to explain the presence

of the solution (0, 0) when the equation a- i °- === c is cleared ofx ’ y

fractions.In the October number of this journal under this discussion the

solution above cited was explained from the standpoint of limits, whichseems to be in accordance with the best usage.We cannot afford to disregard so eminent an authority as Burkhardt,

who makes clear on page 29 in the second part o£ Volume I, secondedition, of his "Funktionentheoretische Vorlesungen,^ a view which isof service in this case. In discussing transformation by reciprocal radii

he explains that s’ == "-is not defined for x === 0 and, in consequence,z

the zero point is an exceptional point of the s plane. He then goeson to state that in mathematics we are accustomed to remove suchexceptions by new conventions; and this he does as follows: "Besides

the complex numbers and their syml)ols already introduced we intro-

duce a new one, infinity, with the syml)ol co , wMcli is to l)e consid-

ered as the result of the division � ."o

Such a statement is sufficient ground for the view our authors have

taken with regard to the meaning of ~- and warrants their acceptanceo

of the solution (0, 0).