214? Prof. Gauss on the Representation of a Surface. 



quantity e*+g*-\-h 2 which is necessarily positive, the sym- 

 metrical quantity resulting from that multiplication, 



abd +gcd -\-haV—ecb ] —gad—hba\ may be applied 

 as a criterion of a similar or reversed position of the parts in 

 the representations 1 and 3. 



In the same manner the similar or reversed position of the 

 parts in the representations 6 and 8 may be proved to depend 



on the positive or negative value of the quantity — ^ = 



CA'-AC AB'-BA' .- . , 

 - = - , or the symmetrical quantity 



EBC' + GCA' + HAB'-ECB'-GAC'-HBA'. 



The comparison of the representations in S and 4* depends 

 on similar principles as that of 2 and 3, and the similar or re- 

 versed position of the parts depends on the positive or nega- 



tive sign of the quantity (Jt ) . (£) - (£ ) . (i? ), and 



in like manner the positive or negative sign of 



/<*P\ /<*Q\ / <*P\ /£Q\ 



\dT /'\dU / \dV J'\dT ) 

 will determine the similar or reversed position of parts in the 

 representations 5 and 6. 



With regard to the comparison of the representations 4 

 and 5, we may refer to the analysis of the 8th article, from 

 which it will be clear that these will be similar or reversed in 

 their smallest parts, according as the first or second solution 

 is adopted, that is, according as either, we have made 



P + *Q z=fp + iq, and P — i Q —f\p — iq\ or 

 P + ,'Q =f(p-iq), and P-iQ =f(p + iq). 

 From all that precedes, we now draw the conclusion that if 

 the representation on the surface, whose equation is $• = 0, is 

 to be in the smallest parts, not only similar, but likewise si- 

 milarly situated to the original in the surface, whose equation 

 is \f/ = 0, we must regard the number of negative quantities 

 which will be found among these four quantities 



ab-baf / d P\/ d 9\ ( d P \ ( d l_\ / dP V^\ 

 h 9 \dt/\du/ \du/'\ dtj' \dT/\dU~/ 



(tu") ( 7t )' — i — ; if tnere 1S none > or an even number 

 of them, the first solution must be taken ; but if there is one, 

 or if there are three negative quantities among them, the se- 

 cond solution must be adopted. By a contrary choice a re- 

 versed similarity will always take place. 



It may besides be demonstrated that if we designate the 



above 



