The core of the strong UV resonance line C IV in the Seyfert 1 galaxy NGC 4151 is characterized by a double-peaked profile whose narrow emission features have relative intensities that vary as a function of the total C IV line strength. We present herein a set of novel correlations between the peak emission of each of these two components and the total luminosity of the C IV line. These correlations are distinctly different for the red and the blue peaks of the double-peaked C IV profile, implying that the physical parameters involved in their formation are quite different. The behavior of the red component appears more in accordance with the behavior of lines observed in other objects, with the peak intensity steadily increasing as the line brightens and broadens (though no specific correlations between these two quantities have, to our knowledge, been previously reported). The blue peak, however, appears much more directly and strongly related to C IV luminosity, and exhibits greater sensitivity to the C IV flux changes, becoming almost extinct when the active nucleus is in its low state. The very presence of these tight correlations, and the very simple, model-independent and direct method employed in obtaining them, imply the existence of specific (hitherto unknown) underlying physical processes. The distinctly different correlations exhibited by the blue and red peaks also suggests that distinct line-forming regions (in velocity space) produce the characteristic double-peaked profile of the C IV line in NGC 4151.