THEORY OF COMPOUND CURVES IN FIELD 

 ENGINEERING 



By Arnold Emch 



In No. 2, Vol. V, pp. 99-108, of the Kansas University Quarterly, 

 which appeared in October, 1896, the writer published the first article on 

 "Compound Curves." Previously, in May, 1896, an article on "A 

 Special Complex of the Second Degree and its Relation with the Pencils 

 of Circles" appeared in No. 5, Vol. Ill, of the American Mathematical 

 Monthly. In this the theorems were established which served as a 

 base for the theory of compound curves. To show the efficiency of this 

 theory and its didactic value in mathematical teaching I published a 

 solution of a well-known problem in field engineering in the Engineering 

 News of December 31, 1903 (Vol. L, No. 27). To this a number 

 of engineers offered criticisms and other solutions in the same journal 

 of March 10, 1904 (Vol. LI, No. 10). On studying these the reader will 

 find, as pointed out by the author in reply to those critics, that their 

 opinions are contradictory in essential parts. The question, of course, 

 is not whether my solution is correct or not; it is in reference to the 

 practical value of the solution. 



Since 1896 I have had many inquiries concerning the theory of com- 

 pound curves and the articles in which my results were published. In 

 view of this, I find it advisable to present all those results once more in 

 a single article, revised and in a condensed form. The mathematical 

 apparatus employed contains nothing beyond common-place mathemati- 

 cal knowledge. 



I. BICIRCULAR QUARTICS PRODUCED BY TWO PROJECTIVE PENCILS 



OF CIRCLES 



The equations of two projective pencils of conies may be written in 



the form: 



U I -\-\U a =o, (1) 



aX+ b TT , . 



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